The Four “M”s of Hirokazu Iwasawa

I first came across the 4M puzzle by Hirokazu Iwasawa (aka Iwahiro) earlier in the year when I met up with a fellow puzzlist in a pub in New York City. The puzzle comprises of a box with a lid and four identical M-shaped pieces which must be placed inside.

I spent a short while trying to pack the various pieces in different ways until I was hit by what puzzlists call the “aha moment”, i.e. a sudden realisation of what needs to be done to solve the puzzle. This is something I rarely experience because I often solve puzzles without realising exactly how and have to backtrack to work out the solution.

That night in NYC we played with quite a few puzzles designed by Iwahiro including some sliding piece puzzles such as the interesting-looking Rectangular Jam, and they were, for me, the most memorable puzzles of the night.

There’s a thin line that puzzle designers walk when setting the difficulty level of their creations. If the solution to the challenge thrown down comes too easy, then there’s little to take away from the experience. There are many very difficult puzzles in all categories that require hours of work and are extremely satisfying to conquer, but there are far less “easy” puzzles in which the solver goes through the same process of struggle and realisation. Good examples are the classic T Puzzle and some of the early creations of Stewart Coffin. Iwahiro’s puzzles fall into this category and perhaps this is partly due to the mathematical basis of many of his ideas.

In late 2014 I chatted with Iwahiro via email about the creation of the 4M puzzle

Saul Symonds: What was the starting point in 4M‘s creation? Did you have the finished concept in mind or did you begin by playing around with different pieces?

Two prototypes of the MMMM puzzle. The pieces are made from corrugated cardboard of two different thicknesses. The boxes are made from MDF and the obstacle, which is on the bottom of the box in these prototypes, is made from wood. (Photo courtesy Iwahiro)

Two prototypes of the MMMM puzzle. The pieces are made from corrugated cardboard of two different thicknesses. The boxes are made from MDF and the obstacle, which is on the bottom of the box in these prototypes, is made from wood. (Photo courtesy Iwahiro)

Iwahiro: I had already realized that the configuration of the four pieces used in 4M is interesting before I thought about designing it. In this sense, the basic design was already in my mind. But the first prototype was a bit different from the present design. In fact, the small obstacle was not on the lid but on the bottom plate of the box. After I played with the prototype, I changed it.

SS: I see on your website you refer to this as the MMMM Puzzle. How did it get the name 4M?

Iwahiro: I myself call it now MMMM when I mention it in writing. (I’m so lazy that I haven’t updated my website for long years, by the way.) But it’s not easy to say “Mmmm”. More precisely, it’s hard to say “Mmmm” so that every listener understands it consists of four m’s. Probably for this reason, my friends started to call it 4M and also I started to call it 4M when I mention it in conversation.

SS: On your website you outline two challenges for the MMMM Puzzle. The first is to put three pieces in the box and then to put four pieces in the box. Did you start with a three-piece packing problem and add a fourth piece, or was it the other way around?

Iwahiro: When MMMM was first published, it had only a four-piece packing problem. Seeing it, my friend, Wil Strijbos, gave me a nice piece of advice that a three-piece packing problem should be added before the main problem.

SS: You seem to use a lot of triangular shapes in your puzzles, for example in the Triangular Jam and Rightangular Jam puzzles. What draws you to this shape?

Iwahiro: I love simple shapes. Triangles may be called the simplest polygons. That’s why.

SS: Many of your tray puzzles and box packing puzzles use some form of coordinated motion. What are the unique difficulties of designing a coordinated motion puzzle?

Iwahiro: In the sense that I come up with very good designs by my standards not frequently, designing is surely not easy. But I don’t know if it should be called “difficulty”. Rather, in my feeling, I need some “luck”. In fact, my designing is not difficult at least in that I could design all of my past puzzles almost only in my head. It’s true that there have been some puzzles that required me to solve some complicated equations on sheets of paper or even with a PC before I designed their details. But those calculations have never forced me to change my original basic ideas.

I have another kind of difficulty in adjustment for manufacturing. But it’s not unique to coordinated motion puzzles.

SS: Many of your puzzles, MMMM included, are suitable for non-puzzlers. They are not overly difficult and only require them to look at the way the pieces fit together in a different way. Here I am also thinking of puzzles like Square in the Bag.

Iwahiro: For me, ideal puzzles are ones that can be solved in 15 minutes by smart non-puzzlers.

SS: You also create logic/mathematics problems.What is the main difference between creating that type of puzzle and a mechanical puzzle?

Iwahiro: The only difference may be in materials used. At least for myself, my mechanical puzzle designs are versions of mathematical puzzles. In fact, I didn’t think I would become a mechanical puzzle designer. But one day I came up with the idea that perhaps I can create mechanical puzzles in a similar way to creating mathematical puzzles. I tried and then I got to know, yes, I can!

SS: How does your work as a mathematics writer impact your puzzle design? (Editor’s note: Iwahiro has authored five mathematics books including two on probability puzzles. He has authored or co-authored numerous other books.)

Iwahiro: I don’t feel there is any big impact to be mentioned. Both are outputs of my mathematical thinking. So there may be a strong relationship between them. But, if I understand myself correctly, it’s not a kind of relation of cause and effect. (Both are effects of some common cause.) (I’m sorry my answer is true but uninteresting.)

SS: Many of your puzzles, MMMM included, are based around mathematical concepts. What are the unique challenges of turning a mathematical concept into a physical puzzle?

Iwahiro: I am not at all a craftsman. And my ideas are often too theoretical. In making a physical puzzle, especially when it’s a mathematical one, consideration of manufacturing accuracy is so crucial that accuracy in the real world often blows theoretical ideas. Indeed, I had got many, many “good” mechanical puzzle designs which turned out to be unrealizable.

MMMM was safe. But the theoretical design of its pieces was much simpler than the real version because I could neglect the thickness of the boards in the theoretical version.

SS: Can you explain the mathematical concept that lies behind the MMMM?

Iwahiro: I feel I should not give a mathematical explanation on my mechanical puzzles as a novelist should not give a literary explanation on his/her novels. For MMMM, I may say only (i) it’s a box packing puzzle with only four identical pieces to be put in, (ii) it’s not too easy nor too difficult for most people, and (iii) most people can experience some ‘aha’ moment with it. (Again, I’m sorry I’m honest and my answer may be uninteresting.)

MMMM is available from Puzzle Master Inc.

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